 We've been talking about the equal sign in the previous couple of videos, so let's go back here and take a look at where we're about to go into, where mathematics we've learned so far, how we're going to apply it to the real world, which is really what the introduction of the equal sign is. It's taking some of the skills that we learn and applying it to wherever we want to apply it. Now, what we've learned so far is just basically simplifying stuff, right? So, we've been going. So, so far right now, we've had, you know, basically used the addition, subtraction, multiplication, division, and the power symbol basically kicking up to another level to, you know, simplify terms, you know, such as... So, we've got a lot of examples and a little more complicated versions of something like, you know, 2a plus 5a, you know, what is that equal to? 7a. Is that on the map? Well, it's not on the map, but it's not very important. So, anyway, so we've done a lot of these simplifications, right? So, we've taken this basic algebra that we've learned so far. What we're about to do is take what we've learned and apply it in the real world, and that's where the equal sign comes in. So, we're about to take a step into an example, introduce a new symbol, which is, as soon as you enter this realm, this is called solving problems. So, there's two types of main problems. You get in the beginning of a high school mathematics. One of them is, you know, you're simplifying stuff, and once you've mastered those skills, you start solving problems, solving equations. And equations, you know, if you do a comparison, sort of like, you know, the English language or any language where you're putting sentences together, right? You're saying one thing is equal to another thing. Why is this important? Because as soon as you start dealing with the equal sign, there's a whole bunch of other symbols that branch off from the equal sign. And you've probably seen a few of these, which is, you know, greater than or less than, less than depending on what you're reading it. You can have an equal to, you can have proportional to, or proportional to, sorry, proportional to approximately equal to, you can have a whole bunch of different symbols. Sometimes you can have, you know, it goes into the branches of set theory and number theory, just geometry, where you use a lot of proportional to geometry, or geometry, you can have two different symbols that interact with each other, right? One is the opposite of either. For example, you have parallel lines and perpendicular lines. When you're trying to say, you know, this line over here is parallel to this line, and that's sort of branching off from the equal sign. And as soon as we enter over here, each one of these has, you know, little specifics that you have to do to be able to use the right circles that they have. And once you're in this realm, you're just simplifying stuff and there is no equal sign. As soon as you have an equal sign, what you're doing is you're solving problems. Solving questions. So what's going to happen in mathematics when you're writing exams or when you're doing things? There's two main types of problems you get. One of them is simplified, which is just crunching numbers down. The other is to solve. And solving, since we've introduced a new symbol, we have to learn how to use this symbol. And for each one of these operations over here, there's certain something you do to be able to move yourself around the equal sign. In general, the way it works is, actually, for the equal sign, 100% it works this way. For the other ones, there's sort of, you know, certain something you have to do to adjust for things, certain properties of symbols. For example, for greater than or less than, if you recall, if you've gone this far, if you multiply or divide by a negative number, you have to flip the signs. The simplest one of these symbols when it comes to solving problems is, again, the equal sign, which is basically the grandfather or where it all started. And then other people came up with new symbols to represent things. The equal sign specifically, I mean, it's obvious what it means. It means one side of the equation is equal to the other side of the equation, right? The other symbols could mean, like, greater than or less than means one side of the equation is bigger than the other side of the equation. Or one side of the equation is smaller than the other side of the equation, or smaller than or equal to. And you use this in a lot of places, right? Sometimes you're not looking for exact answers. You're looking for areas and zones. And we touched upon this, not just a little bit, I think, in geometry, in the geometry and trigonometry sections. But we're really going to delve into it a lot right now and, you know, go through the rules of how to use the equal sign and basically how to do enough problems, hopefully do enough problems to optimize how we use it. Because if you do something when you're moving from one side to the other, if you do it in the wrong way, then your equation is out of balance. And, you know, the equal sign you can sort of think of as a scale. Whatever you do on one side, you have to do to the other side to balance things out. Just a sort of little intro on functions where it takes us is, as soon as we learn the equal symbol and how to work around it, what it allows us to do is create models of anything that we want. And that's the way most of society really works, where you have, you know, financial models, economic models, political models, you know, there's models for everything. And once you learn how to use the equal sign, you know, you no longer have to depend on other people's models to base, you know, either your finances or your life or your education or anything. You can create, you can start creating your own models. You can even start creating your own symbols. Now, there's a whole bunch of these different types of symbols. It's vast. Again, it branches off to a lot of different fields. But all of these symbols were created. I don't know if, you know, one person might have created a whole bunch of these. But certain symbols were created by certain individuals that needed, you know, to express something in mathematics. And they couldn't find the symbol that was appropriate. So they created their own symbols. And there's symbols that group things together, such as, you know, you can do an element of it. What we talked about in the real number, in the real number set that we talked about, where we use this symbol instead as the elements of the natural numbers, the whole numbers, the integers, right? Or one thing we did talk about, which was not the element of. So what you can do is go the element of a certain set. Again, this goes into number theory. And just like we have not the element of like that symbol, we also have something called not equal to. If you don't want something to be equal to something else, the other side of the equation. All of a sudden, just by learning some simple operations, we're going to this side and learning a symbol that branches off to a whole bunch of other symbols, which is just going to give us an immense amount of power to be able to use mathematics wherever we want to use it. We're going towards the realm of applied mathematics. And applied mathematics is us taking certain rules that we learn in the language of mathematics and using it wherever we want. It's like learning words and learning how to use those words. And learning how to take those words and put them in sentences. And that's exactly what the equal sign does. Over here, we put some words together. Over here, we're about to put complete sentences together where we're comparing one side to another. And this also exists. The equal sign also exists in the language of mathematics. And I believe in other languages as well. It's sort of the colon symbol, right? You have the colon symbol in English. And I believe the colon symbol is used to say the same thing two different ways. So something you say on this side, you also say it on this side, but in a different way. Sort of, I guess it's covering two different categories in one shot, right? And you also have variations of this. You have the semicolon in English. This thing in the English language called a semicolon. And I believe that's used for on one side of the equation, you state something and on the other sign you give examples of that something. So I guess this could be something related to set theory, right? With some of the other symbols in the language of mathematics. You could also, I believe, the semicolon you can use for if you state something on one side, you give the negatives of that something. Something that doesn't belong to whatever you say, right? So again, these are just symbols in a language. Just like any other symbols in any other language. There's certain ways to use them and they, you know, to give you more power they allow you to say more things and understand more things. And that's exactly how you should be looking at all these symbols. But what we're going to do start with the equal sign and we're definitely going to look at portionality. We're going to look at the greater than, less than, equal to and stuff like this. Definitely when we get further into geometry and trigonometry and stuff. And you know the set stuff we'll get into, woo! Way, way down the road, okay? Hope this helped. Let's go down the walls or down the, you know down a little ways and find some fresh wall and just do some examples and see how the rules apply when it comes to the equal sign. We want to start using it and moving around from one side to the other.